↑

Properties

Label 152269.55076
Modulus $152269$
Conductor $152269$
Order $52$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(152269, base_ring=CyclotomicField(52)) M = H._module chi = DirichletCharacter(H, M([5,13,34]))
 
Copy content gp:[g,chi] = znchar(Mod(55076, 152269))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("152269.55076");
 

Basic properties

Modulus: \(152269\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(152269\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(52\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: yes
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 152269.pm

\(\chi_{152269}(9014,\cdot)\) \(\chi_{152269}(12992,\cdot)\) \(\chi_{152269}(14539,\cdot)\) \(\chi_{152269}(21926,\cdot)\) \(\chi_{152269}(25147,\cdot)\) \(\chi_{152269}(29440,\cdot)\) \(\chi_{152269}(45037,\cdot)\) \(\chi_{152269}(48446,\cdot)\) \(\chi_{152269}(52772,\cdot)\) \(\chi_{152269}(53308,\cdot)\) \(\chi_{152269}(53435,\cdot)\) \(\chi_{152269}(55076,\cdot)\) \(\chi_{152269}(72756,\cdot)\) \(\chi_{152269}(74209,\cdot)\) \(\chi_{152269}(77397,\cdot)\) \(\chi_{152269}(86237,\cdot)\) \(\chi_{152269}(87563,\cdot)\) \(\chi_{152269}(104580,\cdot)\) \(\chi_{152269}(107580,\cdot)\) \(\chi_{152269}(116956,\cdot)\) \(\chi_{152269}(124155,\cdot)\) \(\chi_{152269}(124376,\cdot)\) \(\chi_{152269}(140161,\cdot)\) \(\chi_{152269}(152222,\cdot)\)

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{52})$
Fixed field: Number field defined by a degree 52 polynomial

Values on generators

\((149567,143313,83318)\) → \((e\left(\frac{5}{52}\right),i,e\left(\frac{17}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 152269 }(55076, a) \) \(-1\)\(1\)\(i\)\(e\left(\frac{15}{52}\right)\)\(-1\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{5}{26}\right)\)\(-i\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{5}{52}\right)\)\(e\left(\frac{15}{26}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 152269 }(55076,a) \;\) at \(\;a = \) e.g. 2